suppose f and g are bounded functions on [a,b] such that f+g is in R[a,b]
Then, does it follow that f and g are also in R[a,b]? i wanto to prove whether it is or not
let f is a continuous, real-valued function on [a,b]
then, for any e, there exist a polygonal function p such that
sup|f(x)-p(x)|<e
using uniform convergence, this might be shown... but i cannot figure it out...
0.9999... = 1
i want to show 0.9999...=1
by showing 0.9999... is a supremum of numbers smaller than 1,
i would be able to prove it.
how can i show 0.999... is a supremum of numbers smaller than 1?
let f(x)
exp(-1/x) for x>0, 0 for x<=0
i want to get f'(x) by using l'hopital's rule, but somehow
i'm applying l'hopital's rule again and again and no clear value is coming out.
i know f'(0) is 0, but i cannot prove it
Let f is monotone increasing, bounded, and differentiable on (a,inf)
Then does it necessarily follow that lim(f'(x),x,inf)=0 ?
It is hard to guess intuitively or imagine a counterexample...
n is a random positive integer.
and how should i show discontinuity when x is rational?
when i prove 'f is continuous when x is irrational', does it follows that
'for rationals, f is not continuous'...?
for f(x)=1/n when x is rational, n is random
so differentiability is not applicable?
since differentiability implies continuity, i tried to use that method..
f: (0,+inf)->R and
f(x) is
0 if x is irrational
1/n if x is rational (n is positive integer)
For each rational and irrational, i want to show continuity/discontinuity of f
Intuitively, i think at each rational f is discontinuous, and at each irrational f is continuous,
but i...